- Open Access
Interacting Dirac magnons in the van der Waals ferromagnet
Phys. Rev. B 112, 094407 – Published 4 September, 2025
DOI: https://doi.org/10.1103/llz4-5k6g
Abstract
We study the effects of magnon-magnon interactions in the two-dimensional van der Waals ferromagnet focusing on its honeycomb lattice structure. Motivated by earlier theoretical predictions of temperature-induced spectral shifts and van Hove singularities in the magnon dispersion [S. S. Pershoguba et al., Phys. Rev. X 8, 011010 (2018)], we go beyond the commonly used thermal magnon approximation by applying second-order perturbation theory in a fully numerical framework. Our analysis uncovers significant deviations from previous analysis: in particular, the predicted singularities are absent, consistent with recent inelastic neutron scattering measurements [S. E. Nikitin et al., Phys. Rev. Lett. 129, 127201 (2022)]. Moreover, we find that the temperature dependence of the renormalized magnon spectrum exhibits a distinct behavior for the optical magnon branch, while retaining behavior for the acoustic or down magnon band. This feature sheds light on the collective dynamics of Dirac magnons and their interactions. We further compare the honeycomb case with a triangular Bravais lattice, relevant for ferromagnetic monolayer , and show that both systems lack singular features while displaying quite distinct thermal trends.
Physics Subject Headings (PhySH)
Article Text
References (60)
- A. K. Geim and K. S. Novoselov, The rise of graphene, Nat. Mater. 6, 183 (2007).
- J. E. Moore, The birth of topological insulators, Nature (London) 464, 194 (2010).
- T. Wehling, A. Black-Schaffer, and A. Balatsky, Dirac materials, Adv. Phys. 63, 1 (2014).
- D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Morozov, A. A. Zhukov, P. Blake, L. A. Ponomarenko, I. V. Grigorieva, K. S. Novoselov, F. Guinea, and A. K. Geim, Dirac cones reshaped by interaction effects in suspended graphene, Nat. Phys. 7, 701 (2011).
- V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, Electron-electron interactions in graphene: Current status and perspectives, Rev. Mod. Phys. 84, 1067 (2012).
- Y. Jin, R. Wang, and H. Xu, Recipe for Dirac phonon states with a quantized valley Berry phase in two-dimensional hexagonal lattices, Nano Lett. 18, 7755 (2018).
- S. Banerjee, J. Fransson, A. M. Black-Schaffer, H. Ågren, and A. V. Balatsky, Granular superconductor in a honeycomb lattice as a realization of bosonic Dirac material, Phys. Rev. B 93, 134502 (2016).
- J. Fransson, A. M. Black-Schaffer, and A. V. Balatsky, Magnon Dirac materials, Phys. Rev. B 94, 075401 (2016).
- S. Bhowmik, S. Banerjee, and A. Saha, Higher-order topological corner and bond-localized modes in magnonic insulators, Phys. Rev. B 109, 104417 (2024).
- P. Di Pietro, M. Ortolani, O. Limaj, A. Di Gaspare, V. Giliberti, F. Giorgianni, M. Brahlek, N. Bansal, N. Koirala, S. Oh, P. Calvani, and S. Lupi, Observation of Dirac plasmons in a topological insulator, Nat. Nanotechnol. 8, 556 (2013).
- P. S. Kumar, I. F. Herbut, and R. Ganesh, Dirac Hamiltonians for bosonic spectra, Phys. Rev. Res. 2, 033035 (2020).
- S. Banerjee, D. S. L. Abergel, H. Ågren, G. Aeppli, and A. V. Balatsky, Interacting Dirac materials, J. Phys.: Condens. Matter 32, 405603 (2020).
- K. S. Burch, D. Mandrus, and J.-G. Park, Magnetism in two-dimensional van der Waals materials, Nature (London) 563, 47 (2018).
- H. Wang, V. Eyert, and U. Schwingenschlögl, Electronic structure and magnetic ordering of the semiconducting chromium trihalides , , and , J. Phys.: Condens. Matter 23, 116003 (2011).
- B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nature (London) 546, 270 (2017).
- D. Soriano, M. I. Katsnelson, and J. Fernández-Rossier, Magnetic two-dimensional chromium trihalides: A theoretical perspective, Nano Lett. 20, 6225 (2020).
- S. Jiang, L. Li, Z. Wang, K. F. Mak, and J. Shan, Controlling magnetism in 2D by electrostatic doping, Nat. Nanotechnol. 13, 549 (2018).
- L. Chen, J.-H. Chung, B. Gao, T. Chen, M. B. Stone, A. I. Kolesnikov, Q. Huang, and P. Dai, Topological spin excitations in honeycomb ferromagnet , Phys. Rev. X 8, 041028 (2018).
- L. Chen, J.-H. Chung, M. B. Stone, A. I. Kolesnikov, B. Winn, V. O. Garlea, D. L. Abernathy, B. Gao, M. Augustin, E. J. G. Santos, and P. Dai, Magnetic field effect on topological spin excitations in , Phys. Rev. X 11, 031047 (2021).
- J. Kim, S. Banerjee, J. Kim, M. Lee, S. Son, J. Kim, T. S. Jung, K. I. Sim, J.-G. Park, and J. H. Kim, Spin and lattice dynamics of the two-dimensional van der Waals ferromagnet , npj Quantum Mater. 9, 55 (2024).
- S. S. Pershoguba, S. Banerjee, J. C. Lashley, J. Park, H. Ågren, G. Aeppli, and A. V. Balatsky, Dirac magnons in honeycomb ferromagnets, Phys. Rev. X 8, 011010 (2018).
- F. J. Dyson, General theory of spin-wave interactions, Phys. Rev. 102, 1217 (1956).
- F. J. Dyson, Thermodynamic behavior of an ideal ferromagnet, Phys. Rev. 102, 1230 (1956).
- W. B. Yelon and R. Silberglitt, Renormalization of large-wave-vector magnons in ferromagnetic studied by inelastic neutron scattering: Spin-wave correlation effects, Phys. Rev. B 4, 2280 (1971).
- E. J. Samuelsen, R. Silberglitt, G. Shirane, and J. P. Remeika, Spin waves in ferromagnetic studied by inelastic neutron scattering, Phys. Rev. B 3, 157 (1971).
- S. E. Nikitin, B. Fåk, K. W. Krämer, T. Fennell, B. Normand, A. M. Läuchli, and C. Rüegg, Thermal evolution of Dirac magnons in the honeycomb ferromagnet , Phys. Rev. Lett. 129, 127201 (2022).
- S. P. Bayrakci, T. Keller, K. Habicht, and B. Keimer, Spin-wave lifetimes throughout the Brillouin zone, Science 312, 1926 (2006).
- S. P. Bayrakci, D. A. Tennant, P. Leininger, T. Keller, M. C. R. Gibson, S. D. Wilson, R. J. Birgeneau, and B. Keimer, Lifetimes of antiferromagnetic magnons in two and three dimensions: Experiment, theory, and numerics, Phys. Rev. Lett. 111, 017204 (2013).
- V. Y. Irkhin, A. A. Katanin, and M. I. Katsnelson, Self-consistent spin-wave theory of layered Heisenberg magnets, Phys. Rev. B 60, 1082 (1999).
- V. Y. Irkhin, A. A. Katanin, and M. I. Katsnelson, On the self-consistent spin-wave theory of frustrated Heisenberg antiferromagnets, J. Phys.: Condens. Matter 4, 5227 (1992).
- V. P. Antropov, M. I. Katsnelson, B. N. Harmon, M. van Schilfgaarde, and D. Kusnezov, Spin dynamics in magnets: Equation of motion and finite temperature effects, Phys. Rev. B 54, 1019 (1996).
- K.-J. Lee, M. Stiles, H.-W. Lee, J.-H. Moon, K.-W. Kim, and S.-W. Lee, Self-consistent calculation of spin transport and magnetization dynamics, Phys. Rep. 531, 89 (2013).
- A. Szilva, Y. Kvashnin, E. A. Stepanov, L. Nordström, O. Eriksson, A. I. Lichtenstein, and M. I. Katsnelson, Quantitative theory of magnetic interactions in solids, Rev. Mod. Phys. 95, 035004 (2023).
- N. D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17, 1133 (1966).
- Y. Liu, L. Zhai, S. Yan, D. Wang, and X. Wan, Magnon-magnon interaction in monolayer , Phys. Rev. B 108, 174425 (2023).
- D. Luo, A. Cong, S. Wen, Z. Yuan, K. Shen, and Y. Liu, Magnon-magnon scattering in monolayer , Phys. Rev. B 111, 094430 (2025).
- Y. A. Kosevich and A. V. Chubukov, Fluctuation corrections to the spectra of one- and two-dimensional Heisenberg magnets, Sov. Phys. JETP 64, 1105 (1986).
- V. A. Alyoshin, V. A. Berezin, and V. A. Tulin, rf susceptibility of single-crystal near the Curie temperature, Phys. Rev. B 56, 719 (1997).
- R. Yadav, L. Xu, M. Pizzochero, J. van den Brink, M. I. Katsnelson, and O. V. Yazyev, Electronic excitations and spin interactions in chromium trihalides from embedded many-body wave functions, npj 2D Mater. Appl. 8, 56 (2024).
- S. W. Jang, M. Y. Jeong, H. Yoon, S. Ryee, and M. J. Han, Microscopic understanding of magnetic interactions in bilayer , Phys. Rev. Mater. 3, 031001(R) (2019).
- O. Besbes, S. Nikolaev, N. Meskini, and I. Solovyev, Microscopic origin of ferromagnetism in the trihalides and , Phys. Rev. B 99, 104432 (2019).
- D. Soriano, A. N. Rudenko, M. I. Katsnelson, and M. Rösner, Environmental screening and ligand-field effects to magnetism in monolayer, npj Comput. Mater. 7, 162 (2021).
- D. Wang and B. Sanyal, Systematic study of monolayer to trilayer : Stacking sequence dependence of electronic structure and magnetism, J. Phys. Chem. C 125, 18467 (2021).
- P. P. Stavropoulos, X. Liu, and H.-Y. Kee, Magnetic anisotropy in spin-3/2 with heavy ligand in honeycomb Mott insulators: Application to , Phys. Rev. Res. 3, 013216 (2021).
- G. L. Stamokostas and G. A. Fiete, Mixing of orbitals in and transition metal oxides, Phys. Rev. B 97, 085150 (2018).
- C. Bacaksiz, D. Šabani, R. M. Menezes, and M. V. Milošević, Distinctive magnetic properties of and monolayers caused by spin-orbit coupling, Phys. Rev. B 103, 125418 (2021).
- Z. Cai, S. Bao, Z.-L. Gu, Y.-P. Gao, Z. Ma, Y. Shangguan, W. Si, Z.-Y. Dong, W. Wang, Y. Wu, D. Lin, J. Wang, K. Ran, S. Li, D. Adroja, X. Xi, S.-L. Yu, X. Wu, J.-X. Li, and J. Wen, Topological magnon insulator spin excitations in the two-dimensional ferromagnet , Phys. Rev. B 104, L020402 (2021).
- C. Tang, Z. Zhang, S. Lai, Q. Tan, and W.-b. Gao, Magnetic proximity effect in van der Waals heterostructures, Adv. Mater. 32, 1908498 (2020).
- A. O. Fumega, S. Blanco-Canosa, H. Babu-Vasili, P. Gargiani, H. Li, J.-S. Zhou, F. Rivadulla, and V. Pardo, Electronic structure and magnetic exchange interactions of Cr-based van der Waals ferromagnets. A comparative study between and , J. Mater. Chem. C 8, 13582 (2020).
- D. P. Kozlenko, O. N. Lis, S. E. Kichanov, E. V. Lukin, N. M. Belozerova, and B. N. Savenko, Spin-induced negative thermal expansion and spin-phonon coupling in van der Waals material , npj Quantum Mater. 6, 19 (2021).
- Y. Zhang, Z.-X. Xie, Y.-Y. Li, S.-Z. Chen, and F. Ning, Improving the Curie temperature of monolayer by Li adsorption: A first-principles study, J. Magn. Magn. Mater. 614, 172686 (2025).
- J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Texts in Applied Mathematics (Springer, New York, 2002).
- K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed. (Wiley, New York, 1989).
- R. W. Wang and D. L. Mills, Spin-wave interactions in the two-dimensional easy-axis Heisenberg ferromagnet, Phys. Rev. B 48, 3792 (1993).
- J. Becker and S. Wessel, Diagnosing fractionalization from the spin dynamics of spin liquids on the kagome lattice by quantum Monte Carlo simulations, Phys. Rev. Lett. 121, 077202 (2018).
- P. Henelius, A. W. Sandvik, C. Timm, and S. M. Girvin, Monte Carlo study of a two-dimensional quantum ferromagnet, Phys. Rev. B 61, 364 (2000).
- V. Zauner-Stauber, L. Vanderstraeten, J. Haegeman, I. P. McCulloch, and F. Verstraete, Topological nature of spinons and holons: Elementary excitations from matrix product states with conserved symmetries, Phys. Rev. B 97, 235155 (2018).
- B. Ponsioen, F. F. Assaad, and P. Corboz, Automatic differentiation applied to excitations with projected entangled pair states, SciPost Phys. 12, 006 (2022).
- Y. Li, Z. Jiang, J. Li, S. Xu, and W. Duan, Magnetic anisotropy of the two-dimensional ferromagnetic insulator , Phys. Rev. B 100, 134438 (2019).
- S. Yang, X. Xu, Y. Zhu, R. Niu, C. Xu, Y. Peng, X. Cheng, X. Jia, Y. Huang, X. Xu, J. Lu, and Y. Ye, Odd-even layer-number effect and layer-dependent magnetic phase diagrams in , Phys. Rev. X 11, 011003 (2021).